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Theorem stoweidlem22 37983
Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem22.8  |-  F/ t
ph
stoweidlem22.9  |-  F/_ t F
stoweidlem22.10  |-  F/_ t G
stoweidlem22.1  |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t )
) )
stoweidlem22.2  |-  I  =  ( t  e.  T  |-> 
-u 1 )
stoweidlem22.3  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
stoweidlem22.4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem22.5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem22.6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem22.7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
Assertion
Ref Expression
stoweidlem22  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Distinct variable groups:    f, g,
t, A    f, F, g    f, G, g    f, I, g    T, f, g, t    ph, f, g    g, L    x, t, A    x, T    ph, x
Allowed substitution hints:    ph( t)    F( x, t)    G( x, t)    H( x, t, f, g)    I( x, t)    L( x, t, f)

Proof of Theorem stoweidlem22
StepHypRef Expression
1 stoweidlem22.8 . . . 4  |-  F/ t
ph
2 stoweidlem22.9 . . . . 5  |-  F/_ t F
32nfel1 2617 . . . 4  |-  F/ t  F  e.  A
4 stoweidlem22.10 . . . . 5  |-  F/_ t G
54nfel1 2617 . . . 4  |-  F/ t  G  e.  A
61, 3, 5nf3an 2024 . . 3  |-  F/ t ( ph  /\  F  e.  A  /\  G  e.  A )
7 simpr 467 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  t  e.  T )
8 simpl1 1017 . . . . . . . . . 10  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ph )
9 stoweidlem22.2 . . . . . . . . . . . 12  |-  I  =  ( t  e.  T  |-> 
-u 1 )
10 neg1rr 10747 . . . . . . . . . . . . 13  |-  -u 1  e.  RR
11 stoweidlem22.7 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
1211stoweidlem4 37965 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -u 1  e.  RR )  ->  (
t  e.  T  |->  -u
1 )  e.  A
)
1310, 12mpan2 682 . . . . . . . . . . . 12  |-  ( ph  ->  ( t  e.  T  |-> 
-u 1 )  e.  A )
149, 13syl5eqel 2544 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  A )
15 eleq1 2528 . . . . . . . . . . . . . . 15  |-  ( f  =  I  ->  (
f  e.  A  <->  I  e.  A ) )
1615anbi2d 715 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  I  e.  A ) ) )
17 feq1 5736 . . . . . . . . . . . . . 14  |-  ( f  =  I  ->  (
f : T --> RR  <->  I : T
--> RR ) )
1816, 17imbi12d 326 . . . . . . . . . . . . 13  |-  ( f  =  I  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  I  e.  A )  ->  I : T --> RR ) ) )
19 stoweidlem22.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
2018, 19vtoclg 3119 . . . . . . . . . . . 12  |-  ( I  e.  A  ->  (
( ph  /\  I  e.  A )  ->  I : T --> RR ) )
2120anabsi7 833 . . . . . . . . . . 11  |-  ( (
ph  /\  I  e.  A )  ->  I : T --> RR )
2214, 21mpdan 679 . . . . . . . . . 10  |-  ( ph  ->  I : T --> RR )
238, 22syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  I : T --> RR )
2423, 7ffvelrnd 6051 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  e.  RR )
25 simpl3 1019 . . . . . . . . 9  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  G  e.  A )
26 eleq1 2528 . . . . . . . . . . . . . . 15  |-  ( f  =  G  ->  (
f  e.  A  <->  G  e.  A ) )
2726anbi2d 715 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  G  e.  A ) ) )
28 feq1 5736 . . . . . . . . . . . . . 14  |-  ( f  =  G  ->  (
f : T --> RR  <->  G : T
--> RR ) )
2927, 28imbi12d 326 . . . . . . . . . . . . 13  |-  ( f  =  G  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  G  e.  A )  ->  G : T --> RR ) ) )
3029, 19vtoclg 3119 . . . . . . . . . . . 12  |-  ( G  e.  A  ->  (
( ph  /\  G  e.  A )  ->  G : T --> RR ) )
3130anabsi7 833 . . . . . . . . . . 11  |-  ( (
ph  /\  G  e.  A )  ->  G : T --> RR )
32313adant3 1034 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  G : T
--> RR )
33 simp3 1016 . . . . . . . . . 10  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  t  e.  T )
3432, 33ffvelrnd 6051 . . . . . . . . 9  |-  ( (
ph  /\  G  e.  A  /\  t  e.  T
)  ->  ( G `  t )  e.  RR )
358, 25, 7, 34syl3anc 1276 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  RR )
3624, 35remulcld 9702 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  e.  RR )
37 stoweidlem22.3 . . . . . . . 8  |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
3837fvmpt2 5985 . . . . . . 7  |-  ( ( t  e.  T  /\  ( ( I `  t )  x.  ( G `  t )
)  e.  RR )  ->  ( L `  t )  =  ( ( I `  t
)  x.  ( G `
 t ) ) )
397, 36, 38syl2anc 671 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  ( ( I `
 t )  x.  ( G `  t
) ) )
409fvmpt2 5985 . . . . . . . . 9  |-  ( ( t  e.  T  /\  -u 1  e.  RR )  ->  ( I `  t )  =  -u
1 )
4110, 40mpan2 682 . . . . . . . 8  |-  ( t  e.  T  ->  (
I `  t )  =  -u 1 )
4241adantl 472 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
I `  t )  =  -u 1 )
4342oveq1d 6335 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( I `  t
)  x.  ( G `
 t ) )  =  ( -u 1  x.  ( G `  t
) ) )
4435recnd 9700 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( G `  t )  e.  CC )
4544mulm1d 10103 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( -u 1  x.  ( G `
 t ) )  =  -u ( G `  t ) )
4639, 43, 453eqtrd 2500 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( L `  t )  =  -u ( G `  t ) )
4746oveq2d 6336 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  ( L `
 t ) )  =  ( ( F `
 t )  + 
-u ( G `  t ) ) )
48 simpl2 1018 . . . . . . . 8  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F  e.  A )
49 eleq1 2528 . . . . . . . . . . . 12  |-  ( f  =  F  ->  (
f  e.  A  <->  F  e.  A ) )
5049anbi2d 715 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  F  e.  A ) ) )
51 feq1 5736 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f : T --> RR  <->  F : T
--> RR ) )
5250, 51imbi12d 326 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  F  e.  A )  ->  F : T --> RR ) ) )
5352, 19vtoclg 3119 . . . . . . . . 9  |-  ( F  e.  A  ->  (
( ph  /\  F  e.  A )  ->  F : T --> RR ) )
5453anabsi7 833 . . . . . . . 8  |-  ( (
ph  /\  F  e.  A )  ->  F : T --> RR )
558, 48, 54syl2anc 671 . . . . . . 7  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  F : T --> RR )
5655, 7ffvelrnd 6051 . . . . . 6  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
5756recnd 9700 . . . . 5  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
5857, 44negsubd 10023 . . . 4  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  +  -u ( G `  t )
)  =  ( ( F `  t )  -  ( G `  t ) ) )
5947, 58eqtr2d 2497 . . 3  |-  ( ( ( ph  /\  F  e.  A  /\  G  e.  A )  /\  t  e.  T )  ->  (
( F `  t
)  -  ( G `
 t ) )  =  ( ( F `
 t )  +  ( L `  t
) ) )
606, 59mpteq2da 4504 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  +  ( L `  t
) ) ) )
61143ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  I  e.  A )
62 nfmpt1 4508 . . . . . . . 8  |-  F/_ t
( t  e.  T  |-> 
-u 1 )
639, 62nfcxfr 2601 . . . . . . 7  |-  F/_ t
I
6463nfeq2 2618 . . . . . 6  |-  F/ t  f  =  I
654nfeq2 2618 . . . . . 6  |-  F/ t  g  =  G
66 stoweidlem22.6 . . . . . 6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
6764, 65, 66stoweidlem6 37967 . . . . 5  |-  ( (
ph  /\  I  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
6861, 67syld3an2 1323 . . . 4  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )  e.  A )
6937, 68syl5eqel 2544 . . 3  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  L  e.  A )
70 stoweidlem22.5 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
71 nfmpt1 4508 . . . . 5  |-  F/_ t
( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t )
) )
7237, 71nfcxfr 2601 . . . 4  |-  F/_ t L
7370, 2, 72stoweidlem8 37969 . . 3  |-  ( (
ph  /\  F  e.  A  /\  L  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7469, 73syld3an3 1321 . 2  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( L `  t ) ) )  e.  A )
7560, 74eqeltrd 2540 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455   F/wnf 1678    e. wcel 1898   F/_wnfc 2590    |-> cmpt 4477   -->wf 5601   ` cfv 5605  (class class class)co 6320   RRcr 9569   1c1 9571    + caddc 9573    x. cmul 9575    - cmin 9891   -ucneg 9892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-po 4777  df-so 4778  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-ltxr 9711  df-sub 9893  df-neg 9894
This theorem is referenced by:  stoweidlem33  37995
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